Variational Optimal Control for Diffusion Guidance

We present a variational control framework that connects sampling-time guidance and reward-based model adaptation through a shared Jacobian pullback. Starting from terminal reward maximization with a path-space KL penalty, the Gibbs variational principle and Doob transform yield a correction to the generative dynamics. A local denoising approximation exposes the pullback that transports output preferences to the controlled variable. This construction gives Variational Optimal Guidance (VOG) in state space and extends to Distilled Variational Optimal Control (DVOC) by amortizing locally improved endpoints into model parameters. The resulting controls share a local variational rule: VOG acts through the denoising Jacobian, while DVOC realizes the parameter pullback by endpoint distillation. Under a linear MMSE model, VOG's exact correction lies in the data subspace. On ImageNet, VOG reduces DiT-XL/2 FID from 2.27 to 2.04 at 250 sampling steps and from 7.58 to 4.44 at 25 steps. After 20 updates on Z-Image-Turbo, DVOC improves HPSv3++ and four held-out evaluator scores over the untrained model, with the highest corresponding means among the trained methods compared. These results connect inference-time control and reward adaptation through a common geometric principle.